On boundedness of divisors computing minimal log discrepancies for surfaces

TL;DR

This paper proves a boundedness result for divisors computing minimal log discrepancies on surfaces, extending Nakamura's conjecture to a broader setting with DCC sets and non-fixed germs.

Contribution

It extends Nakamura's conjecture to non-fixed germs and DCC sets, establishing boundedness of divisors computing minimal log discrepancies for surfaces.

Findings

Nakamura's conjecture holds for surfaces with DCC sets.

Boundedness of $a(E,X,0)$ is established under certain conditions.

Extension of minimal log discrepancy conjectures to more general surface germs.

Abstract

Let $\Gamma$ be a finite set, and $X\ni x$ a fixed klt germ. For any lc germ $(X\ni x,B:=\sum_{i} b_iB_i)$ such that $b_i\in \Gamma$, Nakamura's conjecture, which is equivalent to the ACC conjecture for minimal log discrepancies for fixed germs, predicts that there always exists a prime divisor $E$ over $X\ni x$, such that $a(E,X,B)={\rm{mld}}(X\ni x,B)$, and $a(E,X,0)$ is bounded from above. We extend Nakamura's conjecture to the setting that $X\ni x$ is not necessarily fixed and $\Gamma$ satisfies the DCC, and show it holds for surfaces. We also find some sufficient conditions for the boundedness of $a(E,X,0)$ for any such $E$.